How to reduce the following O.D.E. into first order?
$v v_{yyy} + 2 v v_{yy}-v_{y}v_{yy} + v_{y} v^3 + v_y v -v_{y}^2 = 0$,
where $v=v(y)$ and $v_{y}$ is the derivative w.r.t. $y$. I tried it by integration but I couldn't find the derivative for 2nd and 5th terms.
The procedure is no different from reducing a linear $n$-th order ODE to a linear system of $1$st order ODEs: Write $$ \mathbf{u}=\begin{pmatrix}u_0\\u_1\\u_2\end{pmatrix}=\begin{pmatrix}v\\v_y\\v_{yy}\end{pmatrix} $$ and $\dot{u}=\frac{d}{dy}u\,.$ Then your ODE becomes the system \begin{align} \dot{u}_0&=u_1\,,\\ \dot{u}_1&=u_2\,,\\ 0&=u_0\,\dot{u}_2+2\,u_0\,u_2-u_1\,u_2+u_1\,u_0^3+u_1\,u_0-u_1^2\,. \end{align}